Best proximity point theorems in $b$-metric space satisfying rational contractions

Abstract

In this paper, we prove the existence of best proximity points of cyclic contractions and generalised proximal rational contractions of first and second kind in the setting of complete b-metric space. Some results are also given in the form of corollaries.

Common Fixed Point Theorem for (\(\phi\),\(\mathfrak{F}\))–Multi-Valued Mappings in Cone b-Metric Spaces over Banach Algebra

“…In 1922, Stefan Banach [1] proved a fixed point theorem for contractive mappings in complete metric spaces. In 1969, Nadler [2] Introduce the concept of Multi-valued function. In 1989, Bakhtin [3] introduced the concept of b-metric space opens a gate to further generalization of metric space.…”

“…Reformulate the bmetric space. Many researcher work in this area of research of multivalued function and b-metric spaces [8,9,10,11,12,13,2,14]. Liu and Xu [15] introduced the notion of cone metric space over Banach algebras, replacing Banach spaces by Banach algebras as the underlying spaces of cone metric space.…”